Introduction to Differentiable Manifolds  57430  MATH 536  002
Lecture Time MW 4:00 pm  5:15 pm, Location SMLC 352
Instructor: Dimiter Vassilev Office :
SMLC, Office 326
Email: vassilev@unm.edu
Phone
Number:
505 277 2136
Please note the following information. This page will not be updated beyond January 19, 2024. You should join the class MS Team and follow the information there. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I will send you by email a link to join the class team; please follow it in order to request access. You can find the link to join the team in Canvas as well. Homework, including submission, and the commucation of all course information will rely on MS Teams and OneNote.
Office Hours: MW 1100  1200 (link can be found in MS teams). Feel free to stopby anytime when you have a quick question.
Final exam: TBD (usual room) Please double check the official
Final Examination Schedule.
StudentsExaminations will take place in the same rooms as class meetings unless otherwise announced by the instructor. A change in the final exam day/time must be approved by the instructor's College Dean. Lab exams may be given during the week preceding finals week or during the time period listed below during finals week. It is the student's responsibility to inform their instructors before TBD, if they have conflicts with this exam schedule.
Textbook:
Lee J. M., Introduction to smooth manifolds (Springer, Graduate Texts in Mathematics) Corrections.
Other suggested textbook:
Morita S., Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201).
Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry.
Tu, L., An Introduction to Manifolds.
Please
note the following guidelines for the course:
Description:
Attendance: Attendance at UNM is mandatory, see policy.
Grades:
Accommodation Statement: In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. Accessibility Resources Center (Mesa Vista Hall 2021, 2773506) provides academic support to students who have disabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner.
Homework: Homework is due every Monday at the beginning of the class in MS Teams. I encourage you to work on the homework with your classmates, but you are required to write up your own solutions in your own words. To help the grader, please write your solutions neatly using correct grammar and mathematical notation (no points will be given for work that the reader cannot follow). The ten best homework grades will be used in computing the homework score. Please do not turnin late homework! The syllabus also lists recommended homework problems. These are NOT to be handed in. Work as many as it takes for you to understand the material. You should see me as early and as often as necessary if you are having difficulties with the homework problems .
Schedule and Topics – Spring 2024 (advanced postings
could change)
Class Week 
Topics and Section 
Notes 
1. Jan 15

Topological and smooth manifolds  definitions and examples, paracompactness, partition of unity, smooth structures defined by an atlas. 

2. Jan 22 
Continue smooth structures defined by an atlas  examples. Smooth Maps. Manifolds with boundary. Lie groups. Locally trivial fibrations. Smooth covering maps (HW Pbm). 
See Lee p.268 
3. Jan 29 
Bundle maps and morphisms. Vector bundles. Examples. Diffeomorphisms, smooth partitions of unity. 

4. Feb 5 
Tangent vectors and the tangent space at a point. The tangent bundle. The rank theorem; immersions and submersions. 

5. Feb 12 
Embeddings. Submanifolds regular level set, uniqueness of smooth structure; O(n), SL(n). Tangent space to a submanifold. Defining function of the boundary and regular domains. The Whitney embedding theorem. Transversality. Fiber product theorem for transversal maps. 

6. Feb 19 
Lie groups and algebras. Group actions and the quotient manifold theorem. Free and proper group actions and the manifold M/G. Principle fiber bundles. Homogeneous spaces. 

7. Feb 26 
Vector bundles. Tautological line bundle over the projective space. Constructing new bundles from old. Integrable/ nonintegrable subbundles of the tangent bundle. 

8. Mar 4 


9. Mar 11 
March 1017 Spring Break 

10. Mar 18 
Wednesday, March 20, Midterm Exam. 

11. Mar 25 
The tensor bundle and tensor fields, characterization of (0,q)tensors. Straightening of a vector field near a regular point. 

12. Apr 1 
Proof of Frobenius' theorem. Integral curves and the flow of a vector field. Complete vector fields, 1parameter group of diffeomorphism, infinitesimal generator. 

13. Apr 8 
The Lie derivative of a vector field commuting flows. Derivations of the tensor algebra. The Lie derivative as a derivation of the tensor algebra. Derivations and skewderivations of forms, the exterior derivative. 

14. Apr 15 
The interior multiplication. Cartan's formula. Closed and exact forms, de Rham cohomology and homotopies. Frobenius' theorem in the language of forms (differential ideals). 

15. Apr 22 
Integration of differential forms and Stokes' theorem. MayerVietoris sequence cohomology of spheres. The degree of a map, index of a vector field and Hopf's theorem. 

16. Apr 29 



Final Exam TBD in the usual room. 
